The minimum speed for a blocking problem on the half plane

Alberto Bressan, Tao Wang

Research output: Contribution to journalArticle

11 Citations (Scopus)

Abstract

We consider a blocking problem: fire propagates on a half plane with unit speed in all directions. To block it, a barrier can be constructed in real time, at speed σ. We prove that the fire can be entirely blocked by the wall, in finite time, if and only if σ > 1. The proof relies on a geometric lemma of independent interest. Namely, let K ⊂ R 2 be a compact, simply connected set with smooth boundary. We define d K (x, y) as the minimum length among all paths connecting x with y and remaining inside K. Then d K attains its maximum at a pair of points (over(x, ̄), over(y, ̄)) both on the boundary of K.

Original language English (US) 133-144 12 Journal of Mathematical Analysis and Applications 356 1 https://doi.org/10.1016/j.jmaa.2009.02.039 Published - Aug 1 2009

Half-plane
Fires
Connected Set
Compact Set
Lemma
If and only if
Path
Unit

All Science Journal Classification (ASJC) codes

• Analysis
• Applied Mathematics

Cite this

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title = "The minimum speed for a blocking problem on the half plane",
abstract = "We consider a blocking problem: fire propagates on a half plane with unit speed in all directions. To block it, a barrier can be constructed in real time, at speed σ. We prove that the fire can be entirely blocked by the wall, in finite time, if and only if σ > 1. The proof relies on a geometric lemma of independent interest. Namely, let K ⊂ R 2 be a compact, simply connected set with smooth boundary. We define d K (x, y) as the minimum length among all paths connecting x with y and remaining inside K. Then d K attains its maximum at a pair of points (over(x, ̄), over(y, ̄)) both on the boundary of K.",
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In: Journal of Mathematical Analysis and Applications, Vol. 356, No. 1, 01.08.2009, p. 133-144.

Research output: Contribution to journalArticle

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AU - Wang, Tao

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AB - We consider a blocking problem: fire propagates on a half plane with unit speed in all directions. To block it, a barrier can be constructed in real time, at speed σ. We prove that the fire can be entirely blocked by the wall, in finite time, if and only if σ > 1. The proof relies on a geometric lemma of independent interest. Namely, let K ⊂ R 2 be a compact, simply connected set with smooth boundary. We define d K (x, y) as the minimum length among all paths connecting x with y and remaining inside K. Then d K attains its maximum at a pair of points (over(x, ̄), over(y, ̄)) both on the boundary of K.

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